Normalized Solutions for a Critical Hartree Equation with Perturbation

We study the existence of ground states for the following critical Hartree equation with perturbation -Delta u = lambda u + alpha(I-mu*vertical bar u vertical bar(q))vertical bar u vertical bar(q-2)u + (I-mu*vertical bar u vertical bar(2 mu)* - 2u, in R-N, with prescribed mass integral(RN) u(2) = c(2), where N >= 3, alpha > 0, 0 < mu < N, I-mu is the Riesz potential, lambda is an element of R, and 2N-mu/N < q < 2(mu)* = 2N-mu/N-2. We prove that the critical Hartree equation has normalized ground states and mountain-pass type solutions if it is perturbed by L-2-subcritical part with exponent satisfying 2N-mu/N < q < 2N-mu+2/N Meanwhile,we also prove that the equation has ground states of mountain-passt pe if it is perturbed by L-2-critical or supercritical part with exponent satisfying q >= 2N-mu+2/N.